Binomial Distribution Calculator
Enter n, p, and k to get P(X=k), P(X<=k), P(X>=k), the mean np, and the variance np(1-p), with a distribution chart.
Binomial distribution calculator
The binomial distribution models the number of successes in a fixed number of independent yes/no trials, where each trial succeeds with the same probability. Enter the number of trials n, the success probability p, and a number of successes k, and this calculator returns the exact probability of getting exactly k, at most k, and at least k successes, plus the distribution's mean and variance and a bar chart of the whole distribution.
When does the binomial distribution apply?
A scenario is binomial when it meets four conditions, sometimes called Bernoulli trials: (1) there is a fixed number of trials n; (2) each trial has only two outcomes, "success" or "failure"; (3) the probability of success p is the same on every trial; and (4) the trials are independent of one another. Flipping a coin 20 times, counting defective items in a batch of 100, or counting how many of 12 free throws go in are all binomial.
The formula
The probability of exactly k successes is the binomial probability mass function:
P(X = k) = C(n, k) · pk · (1 − p)n − k
The term C(n, k), read "n choose k," is the binomial coefficient — the number of distinct ways to arrange k successes among n trials. The pk factor is the probability that those k particular trials all succeed, and (1 − p)n − k is the probability the remaining trials all fail. Cumulative probabilities add up the pmf: P(X ≤ k) sums the terms for 0 through k, and P(X ≥ k) equals 1 − P(X ≤ k − 1).
Mean and variance
The binomial distribution has a mean of np and a variance of np(1 − p). So for 10 coin flips with p = 0.5, you expect np = 5 heads on average, with a variance of 2.5 (standard deviation about 1.58). The mean is the balance point of the bar chart; the variance describes how wide the spread of likely outcomes is.
Worked example
For n = 10, p = 0.5, k = 4: C(10, 4) = 210, so P(X = 4) = 210 × 0.54 × 0.56 = 210 / 1024 ≈ 0.2051. The chance of 4 or fewer heads is about 0.3770, and the chance of 4 or more is about 0.8281. As n grows large, the binomial distribution starts to look like a normal distribution — a result known as the normal approximation.
Binomial distribution calculator
The binomial distribution models the number of successes in a fixed number of independent yes/no trials, where each trial succeeds with the same probability. Enter the number of trials n, the success probability p, and a number of successes k, and this calculator returns the exact probability of getting exactly k, at most k, and at least k successes, plus the distribution's mean and variance and a bar chart of the whole distribution.
When does the binomial distribution apply?
A scenario is binomial when it meets four conditions, sometimes called Bernoulli trials: (1) there is a fixed number of trials n; (2) each trial has only two outcomes, "success" or "failure"; (3) the probability of success p is the same on every trial; and (4) the trials are independent of one another. Flipping a coin 20 times, counting defective items in a batch of 100, or counting how many of 12 free throws go in are all binomial.
The formula
The probability of exactly k successes is the binomial probability mass function:
P(X = k) = C(n, k) · pk · (1 − p)n − k
The term C(n, k), read "n choose k," is the binomial coefficient — the number of distinct ways to arrange k successes among n trials. The pk factor is the probability that those k particular trials all succeed, and (1 − p)n − k is the probability the remaining trials all fail. Cumulative probabilities add up the pmf: P(X ≤ k) sums the terms for 0 through k, and P(X ≥ k) equals 1 − P(X ≤ k − 1).
Mean and variance
The binomial distribution has a mean of np and a variance of np(1 − p). So for 10 coin flips with p = 0.5, you expect np = 5 heads on average, with a variance of 2.5 (standard deviation about 1.58). The mean is the balance point of the bar chart; the variance describes how wide the spread of likely outcomes is.
Worked example
For n = 10, p = 0.5, k = 4: C(10, 4) = 210, so P(X = 4) = 210 × 0.54 × 0.56 = 210 / 1024 ≈ 0.2051. The chance of 4 or fewer heads is about 0.3770, and the chance of 4 or more is about 0.8281. As n grows large, the binomial distribution starts to look like a normal distribution — a result known as the normal approximation.
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Frequently Asked Questions
Common questions about the Binomial Distribution Calculator
It is the probability distribution of the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The classic example is counting heads in a set number of coin flips. It is defined by two parameters: n (the number of trials) and p (the probability of success on each trial).
Four conditions, often called Bernoulli trials: a fixed number of trials, exactly two possible outcomes per trial (success or failure), a constant probability of success across all trials, and independence between trials. If any condition fails — for example, drawing cards without replacement changes the probability each draw — a different distribution applies.
Use P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Compute the binomial coefficient C(n, k) for the number of arrangements, multiply by p raised to the number of successes, and by (1 − p) raised to the number of failures. This calculator evaluates that formula exactly and also sums it to give cumulative probabilities.
P(X = k) is the probability of exactly k successes — a single term of the formula. P(X <= k) is the cumulative probability of k or fewer successes, found by adding the terms for 0 through k. P(X >= k) is the probability of k or more, equal to 1 minus P(X <= k − 1). This tool reports all three at once.
The mean (expected number of successes) is np, and the variance is np(1 − p). For example, 100 trials at p = 0.3 have a mean of 30 successes and a variance of 21, giving a standard deviation of about 4.58. These let you summarize the distribution without listing every outcome.
ℹ️ Disclaimer
This tool is provided for informational and educational purposes only. All processing happens entirely in your browser - no data is sent to or stored on our servers. While we strive for accuracy, we make no warranties about the completeness or reliability of results. Use at your own discretion.